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PostulatoryPostulatoryThermodynamicsThermodynamics

Ernő KeszeiLoránd Eötvös UniversityBudapest, Hungary

http://keszei.chem.elte.hu/

• Introduction

• Survey of the laws of classical

thermodynamics

• Postulates of thermodynamics

• Fundamental equations and equations of state

• Equilibrium calculations based on postulates

OutlineOutline

Avant proposAvant propos

Thermodynamics is a funny subject. The first

time you go through it, you don’t understand

it at all. The second time you go through it,

you think you understand it, except for one

or two small points. The third time you go

through it, you know you don’t understand

it, but by that time you are so used to it, it

doesn’t bother you anymore.

Arnold Sommerfeld

Problem with teaching Problem with teaching thermodynamicsthermodynamics

Let’s take an example: probability theory

Postulates of probability theory:

1. The probability of event A is P(A) > 0

2. If A and B are disjoint events, i. e. AB = 0,then P(AB) = P(A) + P(B)

3. For all possible events (the entire sample space S) the equality P(S) = 1 holds

Using these postulates, „all possible theorems” can be proved,i. e., all probability theory problems can be solved.

Important definition: random experiment and its outcome; a (random) event

Fundamentals of (classical) Fundamentals of (classical) thermodynamics: thermodynamics: the lawsthe laws

Two popular textbooks of physical chemistry

Atkins P, de Paula J (2009) Physical Chemistry, 9th edn., Oxford University Press, Oxford

Silbey L J, Alberty R A, Moungi G B (2004)Physical Chemistry, 4th edn., Wiley, New York

(Traditional textbook of MIT; typically, a new co-author replaces an old one at each new edition.)

Definition Definition of aof a thermodynamic thermodynamic systemsystem

Atkins: The system is the part of the world, in whichwe have special interest.The surroundings are where we make our measurements.

Alberty: A thermodynamic system is that partof the physical universe that is under consideration.A system is separated from the rest of the universe by a real orimaginary boundary. The part of the universe outside the boundaryis referred to as surroundings.

(Introduction: Thermodynamics is concerned with equilibrium statesof matter and has nothing to do with time.)

The The Zeroth Zeroth LawLaw of of thermodynamicsthermodynamics

Atkins: If A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then C is also in thermal equilibrium with A. Preceeding statement: Thermal equilibrium is established ifno change of state occurs when two objects A and B arein contact through a diathermic boundary.

Alberty: It is an experimental fact that if system A is inthermal equilibrium with system C, and system Bis also in thermal equilibrium with system C,then A and B are in thermal equilibriumwith each other.Preceeding statement: If two closed systems with fixed volumeare brought together so that they are in thermal contact,changes may take place in the properties of both. Eventually,a state is reached in which there is no further change, and this isthe state of thermal equilibrium.

The The First LawFirst Law of thermodynamics of thermodynamics

Atkins: If we write w for the work done on a system, q for the energy transferred as heat to a system,and ΔU for the resulting change in internal energy,then it follows that

ΔU = q + w

Alberty: If both heat and work are added to the system,

ΔU = q + w

For an infinitesimal change in state

dU = đq + đw

The đ indicates that q and w are not exact differentials.

The The Second LawSecond Law of of thermodynamicsthermodynamics

Atkins: No process is possible, in which the sole resultis the absorption of heat from a reservoir andits complete conversion into work.

(In terms of the entropy:) The entropy of anisolated system increases in the course ofspontaneous change:

ΔStot > 0

where Stot is the total entropy of the systemand its surroundings.

T

đqdS rev

Later (!!): The thermodynamic definition of entropy is based on the expression:

Further on: proof of the entropy being a state function, making use of a Carnot cycle.

The The Second Second LawLaw of of thermodynamicsthermodynamics

Alberty: The second law

in the form we will find most useful:

In this form, the second law provides a criterionfor a spontaneous process, that is, one that canoccur, and can only be reversed by workfrom outside the system.

T

đqdS

Previously: (Analyzing three coupled Carnot-cycles, it is stated that…)

… there is a state function S defined by

T

dqdS rev

The The Third Third LawLaw of thermodynamics of thermodynamicsAtkins: If the entropy of every element in its most stable

state atT = 0 is taken as zero, then every substance

has a positive entropy which atT = 0 may become

zero, and which does become zero for all perfect

crystalline substances, including compounds.

Afterwards: It should also be noted that the Third Law does not state thatentropies are zero at T = 0: it merely implies that all perfect materials have the same entropy at that temperature. As far asthermodynamics is concerned, choosing this common value as zerois then a matter of convenience. The molecular interpretationof entropy, however, implies that S = 0 at T = 0.… The choice S (0) = 0 for perfect crystals will be made from now on.

The The Third Third LawLaw of thermodynamics of thermodynamics

(Péter Esterházy in a novel on communism)

The Kádár-era: filthy land, lying to the marrow, shit as is, in which, aside from this, one could live, aside from the fact that one couldn't put it aside, even though we did put it aside.

Atkins: If the entropy of every element in its most stable

state atT = 0 is taken as zero, then every substance

has a positive entropy which atT = 0 may become

zero, and which does become zero for all perfect

crystalline substances, including compounds.

Afterwards: It should also be noted that the Third Law does not state thatentropies are zero at T = 0: it merely implies that all perfect materials have the same entropy at that temperature. As far asthermodynamics is concerned, choosing this common value as zerois then a matter of convenience. The molecular interpretationof entropy, however, implies that S = 0 at T = 0.… The choice S (0) = 0 for perfect crystals will be made from now on.

The The Third Third LawLaw of thermodynamics of thermodynamics

Alberty: The entropy of each pure element or substancein a perfect crystalline form is zero at absolute zero.

Afterwards : We will see later that statistical mechanics gives a reason to pick this value.

Atkins: If the entropy of every element in its most stable

state atT = 0 is taken as zero, then every substance

has a positive entropy which atT = 0 may become

zero, and which does become zero for all perfect

crystalline substances, including compounds.

Afterwards: It should also be noted that the Third Law does not state thatentropies are zero at T = 0: it merely implies that all perfect materials have the same entropy at that temperature. As far asthermodynamics is concerned, choosing this common value as zerois then a matter of convenience. The molecular interpretationof entropy, however, implies that S = 0 at T = 0.… The choice S (0) = 0 for perfect crystals will be made from now on.

Avant proposAvant propos

After all it seems that Sommerfeld was right…

Thermodynamics is a funny subject. The first

time you go through it, you don’t understand

it at all. The second time you go through it,

you think you understand it, except for one

or two small points. The third time you go

through it, you know you don’t understand

it, but by that time you are so used to it, it

doesn’t bother you anymore.

But thermodynamics is an exact But thermodynamics is an exact science…science…

Development of axiomatic thermodynamics

1878 Josiah Willard GibbsSuggestion to axiomatize chemical thermodinamics

1909 Konstantinos Karathéodori (greek matematician)The first system of postulates (axioms)(heat is not a basic quantity)

1966 László Tisza Generalized Thermodynamics, MIT Press(Collected papers, with some added text)

1985 Herbert B. CallenThermodynamics and an Introductionto Thermostatistics, John Wiley and Sons, New York

1997 Elliott H. Lieb and Jacob YngvasonThe Physics and Mathematics of the Second Law of Thermodynamics(15 mathematically sound but simple axioms)

Fundamentals of Fundamentals of postulatorypostulatory thermodynamicsthermodynamics

An important definition: the thermodynamic system

The objects described by thermodynamics are called thermodynamic systems. These are not simply “the partof the physical universe that is under consideration” (or in whichwe have special interest), rather material bodies having aspecial property; they are in equilibrium.The condition of equilibrium can also be formulated so that thermodynamics is valid for those bodies at rest for which the predictions based on thermodynamic relations coincide with reality (i. e. with experimental results). This is ana posteriori definition; the validity of thermodynamic description can be verified after its actual application.However, thermodynamics offers a valid description for an astonishingly wide variety of matter and phenomena.

PostulatoryPostulatory thermodynamics thermodynamicsA practical simplification: the simple system

Simple systems are pieces of matter that are macroscopically homogeneous and isotropic, electrically uncharged, chemically inert, large enough so that surface effects can be neglected, and they are not acted on by electric, magnetic or gravitational fields.

Postulates will thus be more compact, and these restrictions largely facilitate thermodynamic description without limitations to apply it later to more complicated systems where these limitations are not obeyed. Postulates will be formulated for physical bodies that are homogeneous and isotropic, and their only possibility to interact with the surroundings is mechanical work exerted by volume change, plus thermal and chemical interactions.(Implicitely assumed in the classical treatment as well.)

Postulate 1 Postulate 1 of thermodynamicsof thermodynamics

There exist particular states (called equilibrium states) of simple systems that, macroscopically, are characterized completely by the internal energy U, the volume V, and the amounts of the K chemical

components n1, n2,…, nK . 1. There exist equilibrium states

2. The equilibrium state is unique

3. The equilibrium state has K + 2 degrees of freedom

(in simple systems!)2. The equilibrium state cannot depend on the ”past history” of the system

3. State variables U, V and n1, n2,…, nK determine the state;

their functions f(U, V, n1, n2,… nK) are state functions.

Postulate 2 Postulate 2 of thermodynamicsof thermodynamics

There exists a function (called the entropy and denoted by

S ) of the extensive parameters of any composite system,

defined for all equilibrium states and having the following

property: The values assumed by the extensive

parameters in the absence of an internal constraint are

those that maximize the entropy over the manifold of

constrained equilibrium states.1. Entropy is defined only for equilibrium states.

2. The equilibrium state in an isolated composite system will be the one which has the maximum of entropy.

Definition: composite system: contains at least two subsystemsthe two subsystems are sepatated by a wall (constraint)

Over Over what variables what variables is entropy maximal?is entropy maximal?

isolated cylinder

fixed, impermeable,thermally insulating piston

U α, V α, n α U β, V β, n β

In the absence of an internal constraint, a manifold of different systems can be imagined ; all of them could be realized byre-installing the constraint (“virtual states”).Completely releasing the internal constraint(s) results in a welldetermined state which – over the manifold of virtual states – has the maximum of entropy.

Postulate 3 Postulate 3 of thermodynamicsof thermodynamicsThe entropy of a composite system is additive over the constituent subsystems. The entropy is continuous and differentiable and is a strictly increasing function of the internal energy.

1. S (U, V, n1, n2,… nK ) is an extensive function, i. e., ahomogeneous first order function of its extensive

variables.

2. There exist the derivatives of the entropy function.

3. The entropy function can be inverted with respect to energy:

there exists the function U (S, V, n1, n2,… nK ), whichcan be calculated knowing the entropy function.

4. Knowing the entropy function, any equilibrium state(after any change) can be determined: S = S (U, V, n1, n2,…

nK ) is a fundamental equation of the system.

5. Consequently, its inverse, U = U (S, V, n1, n2,… nK ) contains

equivalent information, thus it is also a fundamental equation.

Postulate 4 Postulate 4 of thermodynamicsof thermodynamics

The entropy of any system is non-negative and vanishes in the

state for which the derivative is zero.

As , this also means that

the entropy is exactly zero at zero temperature.

KnnnVS

U

,...,, 21

TS

U

KnnnV

,...,, 21

The scale of entropy – contrarily to the energy scale –is well determined.

(This makes calculation of chemical equilibrium constants possible.)

(“Residual entropy”: no equilibrium !!)

Summary of the postulatesSummary of the postulates

(Simple) thermodynamic systems can be described by K + 2 extensive variables.

Extensive quantities are their homogeneous linear functions.Derivatives of these functions are homogeneous zero order.

Solving thermodynamic problems can be done using differential- and integral calculus of multivariate functions.

Equilibrium calculations – knowing the fundamental equations – can be reduced to extremum calculations.Postulates together with fundamental equations can be used directly to solve any thermodynamical problems.

Relations of the functions Relations of the functions SS and and UU

S (U, V, n1, n2,… nK) is concave, and a strictly monotonous function of U

S

U

X

S = S 0 p lan e

U = U 0 p lan e

i

Fundamental equations in Fundamental equations in U U and and SS

Equilibrium at constant energy (in an isolated system):

at the maximum of the function S (U, V, n1, n2,…

nK)Equilibrium at constant entropy (in an isentropic system):

at the minimum of the function U (S, V, n1, n2,…

nK)

(In simple systems: isentropic = adiabatic)

To find extrema of the relevant functions, we search for

the zero values of the first order differentials:

K

ii

nVUiUV

dnn

UdV

V

SdU

U

SdS

ij1 ,,,, nn

K

ii

nVSiSV

dnn

UdV

V

UdS

S

UdU

ij1 ,,,, nn

Identifying (first order) derivativesIdentifying (first order) derivatives

We know:at constantS and n (in closed, adiabatic systems):

(This is the volume work.)

K

ii

nVSiSV

dnn

UdV

V

UdS

S

UdU

ij1 ,,,, nn

PdVdU

PV

U

S

n,Similarly:

at constantV and n (in closed, rigid wall systems):(This is the absorbed heat.)

Properties of the derivative confirm:

TdSdU

TS

U

V

n,

at constant S andV (in rigid, adiabatic systems):(This is energy change due to material transport)The relevant derivative is called chemical potential:

iidndU

i

nVSiij

n

U

,,

Identifying (first order) derivativesIdentifying (first order) derivatives

K

ii

nVSiSV

dnn

UdV

V

UdS

S

UdU

ij1 ,,,, nn

PV

U

S

n,

TS

U

V

n,

i

nVSiij

n

U

,,

is negative pressure, is temperature,

is chemical potential.

The total differential

can thus be written (in a simpler notation) as:

K

iiidnPdVTdSdU

1

Fundamental equations and equations of Fundamental equations and equations of statestate

K

iiidnPdVTdSdU

1

K

ii

i dnT

dVT

PdU

TdS

1

1

Equations of state: Equations of state:

),,( nVSTT ),,(11

nVUTT

),,( nVSPP ),,(11

nVUPP

),,( nVSii ),,(11

nVUii

Energy-based fundamental equation: Entropy-based fundamental equation :

U = U(S, V, n) S = S(U, V, n)Its differential form: Its differential form:

Some formal relationsSome formal relations

U = U(S, V, n) is a homogeneous linear function.

According to Euler’s theorem:

K

iiinPVTSU

1

K

iii dnVdPSdT

1

0

Euler equation

K

iiidnPdVTdSdU

1

Gibbs-Duhem equation

We know:

K

ii

nVSiSV

nn

UV

V

US

S

UU

ij1 ,,,, nn

Equilibrium calculationsEquilibrium calculationsisentropic, rigid, closed system

impermeable, initially fixed,thermally isolated piston,

then freely moving, diathermal

S α, V α, n α S β, V β, n β

S α + S β = constant; – dSα = dS β

Consequences of impermeability (piston):

n α = constant; n β = constant → dn α = 0; dn β = 0

V α + V β = constant; – dV α = dV β

Equilibrium condition:

dU= dUα + dU β = 0

U α U 

β

0,,,,

dVV

UdS

S

UdV

V

UdS

S

UdU

nSnVnSnV

0 dVPdVPdSTdSTdU

0 dVPPdSTTdU

Equilibrium: Tα = T β and Pα = P 

β

Equilibrium calculationsEquilibrium calculationsisentropic, rigid, closed system

S α, V α, n α S β, V β, n β

Condition of thermal andmechanical equilibriumin the composit system:

U α U 

β

Tα = T β and Pα = P 

β

4 variables Sα , Vα , S β and V β are to be known at equilibrium.

They can be calculated by solving the 4 equations:

T α (S α, V α, n 

α) = T β (S β, V β, n 

β )

P α (S α, V α, n 

α) = P β (S β, V β, n 

β )

S α + S 

β = S (constant)

V α + V 

β = V (constant)

Equilibrium at constant temperature and Equilibrium at constant temperature and pressurepressure

isentropic, rigid, closed system

T = T r and P = P

r (constants)

S r, V r, n r

T r, P 

r S, V, n

T, P

equilibrium condition:the „internal system” is

closedn r = constant and n  = constant

d (U+U r ) = d U + T r

dS r – P

r dV

r = 0S 

r + S = constant; – dS r = dS

V r + V = constant; – dV r = dV

d (U+U r ) = d U + T r

dS r – P

r dV

r = d U + T r

dS – P r

dV = 0

T = T r and P = P

r d (U+U r ) = d U – TdS + PdV = d (U – TS + PV ) = 0

minimizing U + U r is equivalent to minimizing U – TS + PV

Equilibrium condition at constant temperature and pressure:

minimum of the Gibbs potential G = U – TS + PV

Summary of equilibrium conditionsSummary of equilibrium conditionsVia intensive variables: identity of these variables in all phases φ

Thermal equilibrium: T φ T ,

Mechanical equilibrium: P 

φ P ,

Chemical equilibrium : μ φ μ

i ,

i

Extension is simple for variables characterizing other interactions:

E. g. electrostatic equilibrium: Ψ 

φ Ψ ,

(Ψ 

φ: electric potential of phase φ)

For chemical equilibrium, there is a condition for individual components;for all components that can freely movebetween the subsystems (phases) of a composite system.

Summary of equilibrium conditionsSummary of equilibrium conditions

Other (entropy-like) potential functions can also be applied if needed.

ConstraintsCondition of equilibrium

Mathematical conditionCondition of

stability

U and V constant

maximum of S (U, V, n)

S and V constant

maximum of U (S, V, n)

S and P constant

maximum of H (S, P, n)

T and V constant

maximum of F (T, V, n)

T and P constant

maximum of G (T, P, n)

K

ii

i dnT

dVT

PdU

TdS

1

01

01

K

iiidnPdVTdSdU

01

K

iiidnVdPTdSdH

01

K

iiidnPdVSdTdF

01

K

iiidnVdPSdTdG

02 Sd

02 Ud

02 Hd

02 Fd

02 Gd

Via extensive variables: extrema of these variables in the system

• Postulatory thermodynamics is easy to understand

• Postulates are based on quantities characteristic of the system only

• Relevant quantities (as internal energy and entropy)

are defined in the postulates

• Postulates are ready to use in equilibrium calculations

• Derivation of auxiliary thermodynamic functions (as free energy and Gibbs potential)

is straightforward

• Exact mathematical treatment of equilibria is easy

ConclusionsConclusions

Thus,Thus,

it is worth it is worth

bothboth

teachingteaching and and

learninglearning

postulatorypostulatory

thermodynamithermodynami

cscs !!